Toán học và tuổi trẻ Số 203 (5/1994)

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Toán học và tuổi trẻ Số 203 (5/1994) giới thiệu tới các bạn những nội dung về sử dụng diện tích trong chứng minh; ứng dụng tích phân để chứng minh bất đẳng thức; phép đối xứng trong thiên nhiên. Đặc biệt, thông qua việc giải những bài tập được đưa ra trong tài liệu sẽ giúp các bạn nâng cao kiến thức về Toán học.

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Nội dung Text: Toán học và tuổi trẻ Số 203 (5/1994)

51zor; +1 1994 lrli:ili:tij.rirri:jjri:i:i.rr:t:i:$rj.::rr,tiiill,riiiril::ir,iii+r:ii,ji::iii;:n::t:iit::.ii;riijtl,it :iittr tAp cui na NIcAy rs uANc rH.z(xc ffi *b {}+JS* B&X \#SjJ kd$xsku€ cHg[NC, rJdstf;J Dit THI vao tm Al to ur ., z$ CHI]YIiNI I'O/IN . TIN IIOO ' O ff TOAN Yru I tltlll- lt)$,t -- - -^'\ * tt{\ NIHN SO CAC SO JOSEPHD ' Ufrg 4$nS,ticrr''Phdn' dd, thffifi$1r1ffiimH bet ddng thtic ft )U (O :) -rO I C,(O =' .O, f- -.o Dr :.] n. v1 ; q< o- n!' 1r
- ToAI\ HQC VA TUbI TRE MATHEMATICS AND YOUTH MUC LUC Trang t Ddnh cho cdc ban Trung hoc co sti For Lower secondary school leuel Friends Thdi Vidt Thd.o - Sit dung diQn tich trong chfng urinh 1 Tdns bi€n fip : t Gidi bdi ki trudc NCL]YE,N CANI-I'|OAN Solutions of probl.ems in preuiotts isstte Phd tdng bi4n tdp : C6c bdi tt TU199 ddn T10/199, L1l199, LZllg9 3 NCoDatttr o Gidi tri todn hoc FIOANG C}IIJNG Fun with Mathentatics Binh phuong; GiAi dap bni Bdi torin v6 di6n s6 Nguydn Ngd ; Ai ho gi ? Hod,ng Chnng; D6ng Phtrong Scic cci ddng duoc HOr D6NG elEru rAP : tha tOi kh6ng ? I o Db ru ki ndy Nguy6n CAnh Todn, Hodng Problenrc in this issue Chring, Ng6 Dat Trl, L6 Kh6c Cac bni tt T1/203 ddn T10/203,LU203, L21203 l0 BAo, Nguy6n Huy Doan. o Dd thi vdo l6p 10 chuy6n todn - tin hoc DHTH Nguy6n ViCt Hei, Dinh Quang TP Hd Chi Minh 12 Hio, Nguy6n XuAn Huy, Phan o Tim hidu situ thdm lodn hoc phd th6ng Huy KhAi, Vu Thanh Khidt, Lc Further study of school Maths Hai Khoi, Nguy6n Ven N{au, Nguydn Sinh Nguyen - IJng dung tich phdn dd Khic Hodng L€ Minh, Nguy6n chrlng minh bdt ding thrlc 14 Minh, Trdn Van Nhung, Nguy6n Dang Phdt, Phan o Trd.n Van Vudng - Ci,c s6 Jos6phe 15 Thanh Quang, T+ Hdng o Kdt quA thi hoc sinh gi6i toin qudc Bia 3 QuAng, Dqng Hung Th5ng, Vr1 t Todn hoc vd dbi sdng Drrong Thuy, Trdn Thdnh Mathematics and Life Nguydn Vinh Cq.n - Ph6p ddi xrlng trong Trai, L6 Bn Khdnh Trinh, Ng6 thi6n nhi6n Bia 4 Vi6t Trung, DAng Quan Vi6n. Tru sd tba soqn : 81 Trdn Hung Dqo, Hh NQi DT:260786 Bi€n tQp ud. tri s4: 'rr0 Xttvt THUY 231 Nguy5n Ven Ci{. TP Hd Chi Minh DT:356111 Trinh boy: DoAN ubNc
N lx;i J#, Ti. J ""', *., ii #d" : # _ _Bei toin 2 (D.li XCva) Tr6n c6c canh AB, ta bin ddn crich sfr dung cdc e6ng :, ttrrlc alOi m BC, CA crta \ABC ldy cac di6im C, , At, B'l tudng fng. Chfng t6 rang AAt , BBI , CCI tich. VQn dung nci nhtr-m6t c" khac, "Orig {rhAp ldm phong phri.crich nhin, sily lu6n, vA. AB, CA, BC, qua trinh giAi to6n L,ing rh,i, d6ng quy khi va ctri thi 9y-dd lAm.cho B,C qO=, Knong go bo. Ta dtta vdo cdc khing dinh di bidt : ^,8 . 1) MQt d?.Sr3" chia thinh c6c da giric khOng Gidi :GiA srl cdc dudng thing AAr, giao nhau thi di6n tich da giac bariddu bin! quy tai O. Ap dung cdc ti sd BB t , CC I d6ng tdng di6n tich cd.c da giric d"rroc chia ra. di6n tich ta nhdn duoc : . 2) Hai tam gi6c ddng dang thi t.i sd di6n tich bing binh phtrong ti sd ddng dbng. cA, Sd.o.o BC, 3) Hai tam gidc cci chuns d6v, thi ti sd --= AtB- s*,iit c/: "ou.o s;; va Ci6n tfch bang ti sd 2 drrdng-"uo."' 4) Hai tam gi6c ccj chung dudns cao AB, S*un (dudng cao- bang nhau) thi ti-s6 di6n- tich cBt = s;; b6ng ti s6 hai driy. Nhdn 3 ding thrlc lai vdi nhau ta cd : Ddnh cho cdc ban Trung hoc co sd SU DTING DION TicH Tn,ONG CHUNiG MINH THAI vrdr rsAo Dugi.ddy. s6 trinh bay ^ thrlc di6n tich vdo idcm6t cdng s6 rlng dung vdn d6 : " ,qB1 cAl BCI . a) Ti sd dQ dai cric doan th&ns ------o cci -- the- Bp ArB C,A-'',' Ta chrlngminh di6u ki6n dir : tinh theo ti sd cric di6n tich . Neu BAi to6n 1. Tl,On cdc can-hAc veAB cua MBC ABt CAt BC, ldy^ c6.c didm .B,, C, tuong fng. Goi O h giao .__ Bp AP c,A-' 1 di6'm cia BB, .*/h ' ac; Hay i# o GiA sr] O Ii giao cta L4, udBB,. oB ^. BCt oB, neu = '' vq Ta ki hi6u C, ie CB, grao cta OC vA 4t vi--_R r AB thd rhi rr) di6u ki6n cdn ta c6 ABt 8, ABt CAt BC, AB, CAI BCt ssoc' B,C A,B' CA: ': a,c A,, cr;'do ao Gidi : T:ac6-#+ sor,na BC', BC, xethai rgr6,W,u C;A: ry " va C' le 2 didm chia trong A-B Stanc AC H theo cung 1 ti sd vi vfly C,uit C', trung nhau. h_ norangrancs;;: Tric ld AAt , BB I , CC, ddng quy. Dinh li X6 va Bp= vin {ri1S^khi 41, 8,, Ct nam ngodi cd"c canh ABt + Bp ABt t cia L,ABC. =--a'c =1*P'c-'*F' Blri to6n 3. Cho LABC, BD ld, phAn gidc trong ctla gdc B, hdy chtlng minh \fr, .LBOC vd L,AOC c6 OC chung n6n ti s6 di6n tich bing ti sd dirong cao ha td-i Oa. ;i ;; BD = 2o, ."ouB BC, San.ic BCt a*c 2 oq i BO dci bang : Gidi : Ta cci : "a, "o* d-#: 1B Snsnr, ,AB.BD.sinV saao,,st.,t,rc , BD =s-* "ouo.= s*", Br,; a = \l L +B) smoc- 1 B ,AB.BC sinB 2BC . cos ,
Mat kh6c r Ttrddsuyra:Ro+Rb+Rc> S*ao AD c ,b c, * (; o. ou" ,b nt) ,c *;) o\ Saaac=-*--€ AC a*c -- (;*i) o, (; d. tsD .ttL *y- >- 2Yx,y>o) > 2(do + db +dc) (Std.ungb +- B a*c Blri to6n 6. Gia str ABtD ld 1 trl gi6c 2a . ccts - 2 ldi sao chb drtdng th&ng CD t'i6p xric v6i E drrdne trdn drrdng [inh AB. Chdng minh rang Zac B di6u f,icn cdn va dri dd drrdng th6ng AB tidp a*c Z' xric v6i dudng trbn drrdng [inh CD la cac b) Stl dung t6t mdi lien hO gitra ti sd dien tich dudng th&ng BC va AD song song vdi nhau. vdi ti sd uii drrdng cao, bdn kinh drrdng trbn (V6 dich torin qu6c td ldn th(r 25 (1984)) ndi tidp, ngoai tidp ... cria tam gtrdc, da gi6c. Gid.i: GoiMln Bhi todn 4. Cho 1 da gi6c l6i ngoai ti6p trung didm cria 1 drrdng trdn b6n kinh r. Ta chia da 44c dd AB. thinh &c tam gi6c kh6ng giao nhau Chfng Gqi N li trung rninh ring tdng cde brin kinh dtrdng trtrn n6i didm ctia CD tidp trong cric tam gi6c dci l6n hon r" Mt le hinh Gid.i : Ki hiCu c6c b6n kinh dttdng trdn chidu c.tia M tr€n nQi ti6p trong cdc tam gi{'c dd le CD rt,rr,...,ru. Chu vi cfia chring ldPl, P2, N/ la hinh ..., p,, vd S7, Sr, ..., S,, ll di6n tich ctia chidu cta N tr6n chri-n[. Gqi c}u v1 da gi6c ddu. li P--va diQn AB tich ln S. Ta lu6n cci Pi. P Yi = L , tu - rr AB 2St 2Sz 2Sn Ttgttac6MM,= z * ,z + .... *r'n = * *...* p, , Ta }ai cd O, O, : ZSt -+ 25: 2Su ZS : 1 AB NNr AB.^"NI > pppp +...+------ - (dpcrn). so^,r,rz sINBM = r' 2' = 4 -=r 1 CD: AB,CD Bhi to6n 5. sa"u,'1, = SNT4ND: zMMr' 2 g ?rong tam giric ,{BC ta idy 1 didm M, ki Nhrr vay AD ll BC *BC ll MN vdADllMN hi€u cdc khoAng cdch * Suur*N = SacMN uit, S paglr = S tDuN tit M tdi c6c dinh cira AB..lfNi AB . CD tam grhc la R4, Rg, R6 cdn 48 =- CD kho&ng cdch tdi circ *NN,t2=:-
=r;! = 7a3n = + Lt :@ _j + 1,1:t 3nsd0 nsd9 nsrid Nh{n x6t : Nhi6u ban do khOng nhdn xdt t6t n6n thit qurl nhi6u tnrdng hOp. Cric ban cd loi giAi tdt ld Trhn Phant Nguy€n (9T, NK Hei I*tng),Vuong Vu Thang (9A,, GiAng V6 iI, Ha NOi) ; Ddo L€ Dung (9, Nk Vri Tiru, tfrai Bei T1/199. Tint cd.c chtt sd a, b, c, rl sao Binh) ; Triln Tha,nh Quang (9CT, euEc hoc cho udi ntqi sd tu nhiAn n ta c6 : HuO ; L0 Thanh An (7A, Chuy6n phtic y6n, "". -_q@l.. I-se-Z + t = ( aa- - a + il3 MO Lirrh, Vinh Phf), ... nsria ns6b nsdc nsii.l lE rr rdNc Nsir Ldi giei : Cdch I : Bai T2l199. Cho cuc cla thtlc P(x) uir ek) '1 10t -1 Ki hi6u P,,:17. = thfu ru.an P(x) = Qk) + Q(t - x) Vr € R. * Bidt rd,ng cdc hA sd cia P(x) Ld nhitng s6 thi ld' : 9Pn* l.n nguydn hhdng dnt uit P@) = 0. Tirih P(P(B)). --;;"; Ta cci : aa...q 66--6 H*c cc. + I = Ldi girii : Ctia da sd cdc ban. Tt giA thidt tac6: : dPu . 1g)rt l lUt * cP,, * 1 bP,t . P(0) : Q(0) +Q(1) : 0 v6ix : 0 (i) P(1) = Q(1) +Q(0) v6ir : i = 81a.1,+ e*a+lb)pi, +(a+b*c)p,,+ 1 (1) Tn (1) va (2) suy ra P(l) : g {2) MAt khec : GiA sfr : {W-r3 +l)r= (dP,,*1.tj P(r) = aoft + o1l, - I +. . . * 0,, _ rtc * ar, nsdC trong dci ao,et,...,en lir nhtng sd nguy€n = d.:t4,+sd,2P:t+gd,P,+ I (2) khdng Arn ta cci : Tr) (1) vd (2) suy ra : P(i) = ao * a,. .. 4 art _, * a.,, : 0 81a\, + (l*a + sqp:t+ (o. + b * c)p,, = Sry t& clo = al = ., . = a,, _ I : e, = 0. = d.3P'1, + ad.2Pi + sdP,t th6a mdn v6i vd sd gi6 tri cta p,, _Nghia le P(r) :0 vdi msi r e E. Vdy P(3) : 0. Suy ra P(P(3)) : O. ld'i = 81a I Nhgn x6t : Tdt c8. cdc b4n d6u crj ldi gini 13d':l9a+9b diirng. Cr{c tran sau dAy cd ldi giai fit : Nguy\n [o +6 *t:Scl Kidn Cudng, 9M, Mari Quyri, HA N6i i fe nat Tirclddri Bt*dig+d:9 Long,9T, Lam Son ; Ia Thd Tdn, gCT, NK Thayvdo h€ tr€n tad a = g ;b = g ; c = g IlriQ" YOn ; Trdn Tltanh Son, 9F,, Bim Son, Tcinrlai0.=b=c=d=g lhanh Hda;Vuorug Mai Phuong, 8 torin, NK, HAi.Hung : Dd Dang Tao, gB, Chuy6n V-T, __ Cqch 2 : (Nguydn phi Binh,9A, p?CS Bd ]Inq Hba, Hd Tay ; L€ T6n phat, B"t chuyOn van Dan, He NOi) Le J{hidt ; LA Quang Ndru, B,l chuy6n, Drlc 'DOrg _ -Vi ding thtlc dfng vdi moi n € N n6n dring !_hd, Q$ng Ng6i : Phant Tttdi Hit,0T, vdi n = 2, tttc la : Srrg, Th:ii Binh ; Phan Miruh Nguyet, g{, qabbccal=(dd+l)r(*) !6ng i!{!, Ddng Hdi, QuAng Binh ; Ngiyfin La L4c, BA, Ddm Doi, Minh HAi ; Ddm Khdrth Ndud
I-di giai : Vi di6n tich c5c tarn gi6c ld kh6ng Trdn Vi€t Binh, Nguy1n Cqnh" Nanz, PTCS *Aur n6n ta dqt : D6n"e Da. 74s; thanh Aizn,ffigyydn Ngoc Tdru' 6our r'= Y2 SMMB= z2 Marf Quyri, Ha Noi ; D6 Mi'nli'CdlL 9.lI lhi ' ttdn Nam Sach;Luong Le'Qtm'ng PTNK Hei Ta cri "'SLCMr= : Hrrng ; Phqm'Thdi Hd,DongHqng, Thrli Binh ; AAt x2 + ),2 + z2 :- Dd Dane ?oo, Chuiron Ung ltrda,Tla Tay ; Phont MAt xr Huy Giip,Nang khidu Gia Luong, EIirBic : Dinlt AAi-Mq Itii ttuong, La Thanh An, Chuy-6n-Phfc-Y6n, AM *MAl y2+22 Vinh Phri ;-N guydn Thi Th.cnlt,PTCS Hdng Bdng' =--MAr =-*' H&i Phdng. v[l rlu -t'ttirY BM z2 +x2 Tudng trJ MBt= -V-, BAi T4l199. Ta n6i s6 tU nhiar' n ld s6 c6 cM x2 +y2 tinh chdt P ndu khi n lit udc cia a" '7 udi sd W=-rT nguy€n duong a nis.o d.6 tlt'i n2 cung ld' t16c cila' dt-1 Do dci : a) Chtng ntinh riing nqi sd ngrtyS'n td dbu \fv-;F +V T- {7 +; , \tir+ f co tinh chdt P -xyz f =- i- b\ Chi ra riing tbn tai u0 han h'cp s6 n co S,it dung bdt ding thrJc for+ b7 tlruh chdt P" '# Ldi gi6i (cia nhibu ban) : a) X6t sd nguy€n (o(o - b\' > a) ta dddc td p tny"y. GiA srl o li s6 nguy6n drrong nio dcj 1 .t*z z*t :r*J' ,',a o! - 1i P (i)' Ta cdn chrlng minh V2\ x Y z / al'-l: p-) 1 ,,x )', ,Y 2, rrr lz *;))'- + = -b ((;",-) . (;*t) " (; ThAt viy, tr) (1) +al'-a*a-1i p*a-1i P 1_ >;=.6=312 \z (do at' -ai p theo dlnh li nh6 Ph6ctna), haya=l(modp)+ Khi r = ! : z ta cd P Cai gi6 tri nh6 nh{t vd Pu,1n= 3{,'Lric d Nhin x6t: (a -.1)(d' - I + at' - 2 1. .. * a * 1) = 0 Da s6 sic ban giAi nhu tr6n' (nrodp2) hay aP - 1 i P2. (Dicm). b) X6t hgP sd ru : P t.P: (*), vdiP, , P2ld hai Nhi6u ban ghi kdt qu6 la 3 tl8. sd nguy€n t6 kh6c nhau tiry y' Gia srl a Ii sd Cric ban giAi A rrg.ryl, dtrong ndo dd md tdt bai niry : A ay - 7 i n , hay oPttrt - 1i pt pz. Khi dd : Nguydn LA L{c, cdp II Ddm Doi, ^ \''' 'trg, ,/ "'A (al'tl'r - I i Pt (2) vd (aPtf'z - 1 i p2(3)' Theo \ MinhHeit/r'\i1'... kdt que ciia phdn a), til (D vi (3) + Nguydn. Dic i \ @rY; - 1i pi (4) va (at il'z - t t P3 ;:2't}._ (5)' Ma Tudng PT i ,.t ti -.,\ chuy6n Gia /.." \, ']' fl ua pj nguyon td cirng nhau n6n ti (4) va (5) Lai ; LA "i.'n'"" Trdn k:*-* -^-.-.-u*"-----Y At c ta duoc o\tPz - tr i p2t .n), hay dt - 1 i n2 ' Thd ;;:; DuY, fr Al Luong Le Til, Di6u niy chfing t6 n co binh chdt P' DO thdy crj v6 han frqp s6 dang (*). VAy cci v6 han hgp chuy6n LO Khidt, LA Quang Ndnr, Chuy6n Drlc sd cri tinh chdt P. (Dpcm)' Phd-, QuAng Ngai ; Dd.nt Khd.nh Hita, Vd Thi Ly, Chuy6n Lrrong Van Ch6nh, Phri Y6n ; Tr6'tt NhQn x6t. 1. Cd fiL nhi6u ban grli ldi giai fni rui" Thanh, CII D6ng Mf, Ddng Hdi, cho bii to6n. M6t sd ban do khdng bidt v4n Qu&ng Binh ; Vi€n Ng.oc Quang, PTC'S Ba dung kdt quA cta phdn a) n6n giAi-phdn b) dirh,"Thurrh Hcia ; Trd.n Dic Quybn, PTCS Xfrong ng6n gon. Nhi6u ban kh6c, do kh6ng Trdn DEng Ninh, Nam Dinh, Nam }{.d; Dod'n hidu dring dinh nghia "sd c6 tinh chdt P" n6n Dinh Truig, Nguydn 8,6. Hilrtg, Nguydn' Anh giAi sai phdn b). Tti, Nguydi Hadng Duorug, Trd,n Bd'ng,Ng-uydn 2. Cdcban c
?ti., Nguydn Hodng Duong, PTCS Trtrng Til (2), v6i z dri ldn thi x2rt + 1 < xr,, _ , (do Vuong), Nguydn Phn Qudng, Pham. La Hi"rn[, 0 < a < l), di6u niy mdu thudn vdi giA thidt PTCS Trung Nhi, Nguydn Phn Binh PTCS 86 ddy x, =+ *oo khi n n a. Van DAn), D6 IIb Ngo, pTCS D6ngDa) ; Mai Thanh Binh - 7M, PTDL Marie Curie, Hi Ydy a > 7. Tt (1) ta e 0. Vinlr Phrl) ; Nguydn Trqrug HQu, phant Dinh Chinh, T6 DOng Vit (CT DHTH Ha Ngi) ; Dinh Kdt luAn : a Z 1, B > 0 tiryy. Trung Hdng, Nguydn Tudn Hdi, Nguydn Thai Nh6.n x6t : Cric ban grli ldi giei v6 tda soan Hd (PTDL Marie Curie, Ha NOi) ; Trinh Thd d6u cd ldi giAi dung. Huynh, Truong Thxong Hoitng - (PTTI{ L6 Hdng Phong, Nam Uil ; La Minh Hidu, L€ NGLIYEN VAN VAU Trqng Giang, Nguydn Duy Hi^tng, Hodng Ngoc BAi T6/199. Cho sd nguyAn d.uong ld k ud. TAm (PTTH Lam Sdn, Thanh Hda) ; phitng sd nguy1n duong n thfu ntdn k < n. Gid sl c6 San Ld.m (PTTH Phan tsOi Ch6u, NghQ An) ; b0 sd c6 thi tt! (x, , x2, . . . ,nn) Nguydn Thanh Hdi (PTTH Dio Duy Tr), udi xie { -t; +;} V i =T;i. Cho phip th4c QuAng Binh) ; Ddo thi ThiLn Huortg, io Van Thdo, Hb Si fhdi (PTTI{ I)6ng Hi, euAng hien phdp tod.n sau : ldy h s6, nilm d h ui tri hhd.c nhau, cia b6 sd {x1 , x2, . . . , xrr) r6i thay Iri) ; Vo ViQt Hilng (PTTI{ }Ioing Hoa Th6m, Da Ning) ; La Anh Yl? (Qu6c hoc HuO vd nfii sd bdi sd rldi cia n6. ll_quydn Cdng Tudn (PTTFI Trtlng Vrrong, euy Cho hai bQ sd c6 thtt ttl : Nhon). A = (al, a2,.. ., a,,) udi a,e { -1 ; +l} ructrYiN fHAt: fr4tNtt Vi=T,n BAi TSli.B$. V6i cdc s6 th1c d,uong B: (bl,b2,...,brr) uoi b, e { -1; +;} xo, lu, a , B ta xet ha.t. day sd {*u} ,, o Vi:T,n ud tlu) ,, r,, doo, xdc dinh ruhu sau Chtng minh riing nhis ui€c tttuc hi1n li€n r,, tidp m|t sd hitu han liin ph€p thay s6 n6i tr€n {l*n*l=aY,,413 VrL = 0, l, Z, ... (1) ddi udi bO sd A ta cd thd nhQn d,uoc b0 sd B. lln*t:axr,*PIY,,' t Ldi giai kiaPhirng Son LOnt., PTTH phan T)m dibu kiAn ciln uit d,i ddi udi a , p dd ta Bdi ChAu, Ngh6 An, V6 Qudc Hilng vit. ph.qnt c6 xn-*a ud./r,*** khi n--+*co u6i ntgi COng ThiAu, tnldng Chuy6n IG Khidt, euAng xo,Yu>O Ngei). Trudc hdt, ta chrlng minh : Nh{n x6t : Nhd vi6c thrrc hi6n li6n tidp phdp Ldi gi6i (cria da sd c6c ban) : tod.n cria d6 bei ta cri thd ddi m6t sd o, bdt ki cria GiA sfi xn-** !rr+*a. , A thAnh sd ddi cria nci, cdn (n - 1) sd cdn lai cria A vdn giu nguy€n giri tri cria minh. X6t O < (t < 7. Dat xolln: c. Tr) gie thi6t (1) ta suy ra Thdt vdy, x6t a,bdt ki thu6cA. Ldy ft sd ndo : d
n-k- I s6 cbn l4i (ngodi ai,dir,...,ai.r) 62 4"2 -o2 eosd = cira A crlng gifi nguy6n gi6 tr! cria minh 2b" lcosa : cosA (do kh6n9b! ph6p to:in tdc d6ng t6i)' d - b) Nh.Qn *Zt Auqc ch.rlng minh' Vi : cosp =T*1:,.:!i::::, + c2 a.i ;6ie{ -r; +.t} v i = 17 non v6i m6i I o2 +f -") ( 7, 2, ..., rt, cilc sd a;, b, ho4c nhtr nhau ho4c cogl = ' zdr) chi khac nhau v6 ddu. Do dd, tt NhQn xet d +il +c2 tr6n d6 ddng cri dddc di6u dd bei y6u cdu Viy max F(x,y,z\=--nabc ' dat chring minh. khi * : :; -' , Nh$n x6t : Trong sd eac ban grli ldi giai drroc, ching h4n. | - A, Y cho bii to6n, chi cri mQt ban cho ldi giAi sai' - -ft -t- z--v. Nhi6u bpn cho ldi giei khOng ng6u gon' Ngoni 2 c5c ban dd n6u t6n d tr6n, c6c ban sau c0ng cti NhAn x6t: Hdu hdt cric ldi giei grli ddn ddu ldi giei 6t : Dd Hb IlSa (PTCS Don-g- Da, Hi dring chi cd mOt s6 it rrdc lttgng bdt d6ng thrlc NOiJ, Nguydn Pkrt Binh ( PTCS Bd !-an Dirn, khOig ch4t, vi vay cho k6t quA sai' Ngon! cich Ha NOi) i Nguydn Thdi HiL ( PTDL Marie giai t""ao,'mQt sE- bp cbn cho crich giAi hinh Curie, Ha NOi) ; Nhfr Qui Tho, Nguydrt Duy [9c d1ra. tr6n bidn ddi v6cto. Hitng, Phqm Minh Tuan (PTTH Lam Son - NGUYEN VAU UAU Thanh Hda) ; Dod.n Trd.n Trung (PTTH Phan' BQi ChAu, NghQ An) ; NguYdn Thanh Hdi ( BAri T8i 199 , Gid. sit H , Au , Bu , Cn lan luot PTTH Ddo Duy Tt, QuAng Binh) vn L€ Anh ld truc td,m., trung d.idnt cia cdc doart' BC, CA, Vr7 (Qu6c hoc Hud). AB cia tant gidi hh1ng cdn ABc. Cdc dudng . th&.ne tan luot qua A, B, C uit uuing g6c udi NGUYEN KiI,{C MINTI HAo: IIB,r, HCri cat ctu dudng thang BC, CA, C, . Chrtng ninh rarug Bai T7l199. Cho a, b, c lit. d0 dili 3 cqnh AB lhn luqt tqi At ,81, cia ntQt tant gid.c. Xet cd.c sd x, y, z tban nt'an At,81, C, thang hd.ng. x*y*r=U. Lbi giii: (dUa theo Tim gid. tri l6n nhd,t cila bidu thuc : Nguydn Thai sinx sinn,t sinz FId. 1OM, PTDL F(x,Y,z)=;+f * Marie Curie, Hd " Nqi). Gsi Ldi giai (cria da s6 cdc ban) A2,f_2, C2 ldn Ao Azc At f)atr : y = - p, z =| -r thi lddt la chAn ; -", ; dtrbng cao h4 tr) a .+ p +y = xt uit sinx = cosa, siny - cos?, cric dinh A, B, C. Do AABC khOng cdn n0n SLLZ = cosy A,#A,,. QualI kefia!7x songsong v6iBC,ta St dqng ddng nhdt thr-ilc. cd chum H(x,AoBC) la chirm di6u hba' Anh cria Z(bccosa * catosB * abcosY) = chim.H(rAu BC) tn:ngphdpquay(H i 9Oo) li m6t = a2 *b2 +c2 -@sinB -bsina)2 chirnr v6i cdc tia ttldng rlng song song vdi eirc t'ia -{bcosa *acosP -c.7) cita chirm A(A2A..CB). Suy ra chim ndy c(tng s6 c
Ap dqng dinh li Menelauyt ta cci dpcm. .hi€"n nhi6n yr)2 + (yz -xz)2 + (xy -xz)2 > 0 - \*y . Nhgn x6t : Cci 42ban giei bai niy. Trr) m6t = d[ng. sd bqn kh6ng x6t hti6ng md dd kdt luAn Ding thfc xey rB . 2Sr{S Pz+Ps ' pt *pt-'t pt tp:*t' \guydn Thanh Hdi (72T, Dd.o Duy Tr), QuAng vdi p1, p2, p j li 3 sd duong bdt ki. Binh) ... dd srl dung lrrong giric drra v6 chring minh bdt ding thrlc trong tam giric ABC li : lqi grAi : Ap dsng Ller cl6ng thtic BtrnhiacOpski ta cci ABC +tg tgZ +tg > rJS : 2 , . , o2t o2' a2, Mdt s6 b3n dd rnp dung bdt ding thrlc LPZ+Ps ltt+ps p1tp,J Tr6brrsep de coi a1t a2, a, ld binh ding vd x [(pz+p_.) +- @ t + p) + @,+p)] > P1, P2, P j le binh d&ng dd g)a srl at 7 a2 2 a, si1 p t > p t > pj.Cdcban nh6cho >- (a., *(r. la.';: khi o, > 0.)> a, thi p1,p2,p3 dAu cbn binh ^, aj *j a2, ding nrra ! '*2 (rr.ft =;;*or" o;or) @t+Pz+P) > r-ErHdNc NuAl > (at t a, * ur'12 B}ri T10/199 Gei G ld trqng td.nt trt di1n P1 )P:)P-t, S.ABC. M ld. mOt didnt. tltuQc LABC. Cdc duimg :-_--=- a7 i ----:-- ta-t +ps t pt +p:*J Qa 2 thdng qua M song song u6i GA, GB, GC the-o -t- Pt +Ps ' pt thu tU cdt cdc ntet (GBC), (_GCA), (GAB) tqi A1, 1 - B t, C t Chilng minh rdng GM di qua trang tdm L ArBtCt. Ta chi cdn chrlng minh Ldi giii : Trrt6c hdt, LABC ld tAp hop didm 1_ thudc it nhdt m6t trong ha doan thing AB, BC, j[(a, +a r+a 3)2 -2 1o] +ai*"rr)l > 2s\i5 Clt n6n dd bai cdn duoc srla lai hay r+a r+a j)2 - (a 1oi+oi*"3), 4S\1, (*) Z ld : M thu6c Thdt viy: (*) o mi6n tam giric *ai -@ 2-ct )2 +a22-@ 1 --a i2 +a23 - (a ;a2)2 ABC. Ta s6 > 4S{5 chrlng minh bii, +e (at - a)(at * a, - ar) * az + * (.a.t * a, - ar)(.a, - a, * ar) * torin tdng qudt trong dd G li ,*i,,D m6t didm bdt ki * (o, * a, - ar)(a, - ar * ar) > 4,{i S (*n) kh6n.g thu6c mf @tsc) a,*a-*o,, Dat o GqiN lA giao didm cta mf (MArBr) v6i GC. -*ey- 2 -;p-ai=x; Do MA,llGA vd MB,llGB (1) n6n NB,l lMA, vir P -aZ= Y; P -03 =J z thi NAtllMB; hay MA,fuB, la hinh bt'nh hdnh. (**) *o Suy ra IufN, AtBt dc; tring didm chung, goi ld e+4(xy * yz * zx) > 4{g \G+J +:,l[{rz didm 1 (2). M+t khdc, do MCt ll GC n6n C, e (*y)2 + (yr)2 + (zx)2 + 2(xy2 z * # ye + xyz2\ > mp(MGC), suy. ra GCI le giao tuydn cria hai o- 3(xy2 z * xyz2-+ x2 yz) nr6t ph&ng (MGC) vA (GAB). T,] tr6n, ta lai cei
MN la giao tuydn ctia hai m4t ph&ng MGC) Bei Lzlr99. va (MArBr). Hbn nta, do (1) n6n rnf GAB) ll Cho mach tliQn mf (MA,B,) sr-ryraMN ll GCt.MdMC,llGN (St) ruhu linh uE. Trong d6 antpe n6n trl gtde MC rGN ln hinh binh hinh, do dci kd A) ctti 0,1A, MG di qua trung didm J cira C,N, Kdt hop u6n kd rL) chi didu nly vdi (2), ta cci giao didm O cria C,Iv6i 12V. Ndu ddi ch6 uOn kd ud MJ ld,trong tAm A MNCt*t = ru o #, f,, antpe hd thi khi dct ampe kd chi ld trgng tAm cia A AtBlcl, dpcm. 0,02A. Tinh gid fti crta di€n trd R, bidt UAB Nh{n x6t : Cd 34 ban tham gia giei bai hh6ng ddi. ndy, trong sd dci crj 4 ban d6 xudt va giAi bni Ldi gi6i : Goi R, vd' Ra ld di6n tro ctia v6n to6n tdng qu6t nhu tr6n. Cric ban sau day crf ldi giai t6t: Nguy\ru Tudn Hdi (11M - Marie kd vi ampe kd. Lric ddu ta cri : Curie, He Noi), Dinh Thdnh Trung (10A.. nAC Uoc. BR,,7in +/rr) oiirh Ht N6ii' ; va rhi.ntt Lons (tdp rrcf -R; trrldng PTNK HAi Hung) ; La Qudc Trinh ucn Rt (i0T - Lam Son, Thanh Hria). u,lc RUan nANc vtBN (1) ' Ry = norq* rypU * BAi L1l199. MOt bintt. c6 tttd tich V d,uoc ndi udi ntOt bont hilt kh; (bonr tay). Durug tlclt Khi ddi ch6 (A) vn ff) cho nhau ta lai cd RRAUAT' t6i da mdi td.n bont.lit.Uu: *r.Hoi ptdi bont u'A('= RoRv+R(RA+RL) vdi cudng dd hil t6i thidu bao nhi€u ld.n dd dp sudt d trong binh thdp hon Sntnt Hg. Cho rang dp sudt ban dbng diQn qua ampe kd Id ddu eila binh lit 760ntm. Hg uit nhi€t d6 hhdng u'or' RU,,Ig thoy ddi trong qua trinh bont. Q) Ldi giai : GiA stl sau m6i ldn troin thd iich ^A R,,t R,tRv + R(RA + Rv) khi trong binh dtroc gi6n nd ti V ddn V +V, Kdt hop (1) va (2) ta co Vi nhiet d6 trong quri trinh bom khOng thay uoc un, ddi n6n ddy ln qud trinh d&ng nhi6t PV : const. 12 Vdi ldn bom thrl nhdt P,(V*V1)=PV- I'o = n, -Rv = ,\ o,g2 = 60oo ,Pt : PV Vdi ldn bon thf hai uuc t2 frq vir J? : : : 150 f2 uoC t2 h) = PrV-P, PV. IA _R, o'1 - --;^ P2(V + = , +V1r)z !-;. Tudng tu, ooo ' \V vdi ldn bom thrl n : Nh{n x6t : Cd 65 em de grri bdi gi6i, trong dci cci 60 em de crj kdt quA dring ; trong sd dci D PVI P dec biet hoan ngh6nh 23 em l6p 9 da cd ddp s6 - (v +vt),t ' rt -_ (1 + o,os),' df ng (cci em de gti t6i cL2 cdch giAi cria minh). Cdc em sau dAy dd cci bii giAi gon vdt6t : LA Trong d
Girii ddp bdi BAI TOAN VE orErr s6 I GiA sit bAng sd da didn nhu sau : vd theo giA thidt thi A tI (' AI}G ABC D E r +CFI vd,*GHI o6xc PErroNt; s6c c6 oAxc I G II I EEE DEV DUgc rEA ror rs0xc r vdD+E+F:B+E+H=L8 Trong s6ch Cd hoc tinh hoa (NXB T16, 1992) cd mdu chuy6n sau dAy =E: I : f}i6,u nAy chi cd thd xAy ra n€u D +tr' i ho4cD +F' = 14 viE = 4. Qua m6t vii md NhAn trung dhi sdng lAu 1' m6rn thay chit bing sd ta thdy chi cd khA nnng MQt h6m vua Vri Dd nhn Hrin n
Tim nrl tri nh6 nhdt cria tdng cric s6 dtrgc vidt trong bAng. Ncfi UAN Bei T9/203 : Tr6n in4t phing cho didm / cd dinh. Ba drrdng tron (Oi ,Rr), (O2,R2), (O3 , ft3) cilng di qua 1. Ngoni ta O, cit 02 t4i C , azcilt OTtar A , 03 cat O t t4i B. Bidt ring I nim d trnong tam g16c ABC. MOt drJdng tt-r&ng C6c l6p PTCS MNtidpxricv6i (O1), (Or) ldn lnatAM, N, kh6ng c1tt (O3); tudng tu, dudng th&ngPQ tidp xric v6i Bni T1/203 : Cho n s6 thrlc a,a2,...,a,, (O), (O3) vd dridng thing EF ti6p xric v6i (O:), thu6c doan [-1. 1l sao cho al + a) i... + a] : O. Chrlpg minh rdng (Ot). Gie sit cric drrdng trbn (O1), (Or) vd (or) n tlray ddi.sao cho al + al + R3 < 3. ar*ar+...+or=B- - a) Hiv tinh bdn kinh cria c6c dudng trbn vir NcuvEn oE, khoinecach t6rn c'ic drrbngtrbn do, sao cho tdng ElAi T2l2O3 : Phdn tich da thrlc ?-s = atuuN + dtNPQ + &^JEF ldn nhdt. .rB -'9tlt'1 + "! tirhnh nhan ttt & ) Chfns minh rdns khi L'5 dat (lt/dc gla t u+t.ugep-s,A-q4 vO II[,LI BiNFI irnat thi di6n tich" luc gi6c ldi MNPQE Bari ?3/f;S:i : (Jlw do?n th&ng Ats, didm C |#fffif"Tl;li:fll:FT#r{:}iry'$F6r} cringr dat dugc tri l6n Ei6 tr! duoc gie nhEt cta I6n nh*dt ctia nci nci. GIANG :rim gita A, "& ri l;ir (r u6ng gcic v6i AB. TRINI-I BANC GIAT Tr6n-tia Cx rt'";,' 2 dielrr D; E sao cho Bai Tl0/203 : GiA st trl di6n *BCD c6 tdt CE CA cL iic canh iidn xric v6i m6t mai cdu ndo dd. Gie iluungtdrn ng:';;i ti€ip ;\.lDCcat sr} tidn didm cira mdt cdu'v6i ddc canh AB, AC, A = * = {3. AD. cD, DB, BC th'eo thf tu la M N, P, Q, R, drtdng trdn ngoai tiSp A BEC tai H (H * C). S. Chrlng minh rang MQ, NR, PS d6ng quy. Chtlne nrinh rang H(: ludn lu6n di qua tn6t NGT]YEN MINH HA didm co'dinh khi f- r'hurdn dOng tr6h doan thineA.B.'Bai todn "r',tr ;lung kh6ng khi thay Cde el6 Vat f {3 b6i nt cho trttdc (nr. :' 0t BAi {,i/203 : Yat lul kh6i lugng lkg ctdgc NGUYEN DUC TAN girr tr6n mat AB ' l'n,' mdt sgi day, mat A-8 f,ghicng 30' so vr,r phuong nanr ngang, -M4t C6c l6p PTTH AB quCu xung quanl' irue ihAngrdrtigAl{ vdi Bhi T4l203 : Giai bdt phrrong trinh vin tdc gdc kh6ng ddi u., = 10;- DdY bi d6n x2+Gx+z
'N71203. Find all functions : f ;,R \*-11 +::B satisfying the condition ^,4x-2, f(,+l)=2f(x\Vr;'-1 NGI.IYE,N VAN MAI.I For Lower Secondary Schottls n real numbers T'I.1201', Given T8i203. A given square board consists of a1ta>..r an from segment [-i, 1] and 1994 x 1994 equal squares. Each square is al + a3, +...+ af, = O. Prove that labelled by a non-negative integer satisfying the following condition : If a certain square is a, * a, +.".+ a,,
oE rm UAo lfip t0 GHUYET IOAU - TlIl HgG IggS-I9gr Truolrs oal h0c rfig rrqp T'P Hd clrt lulinh - vdNc t ucAY zrlslrsss,triot cIAN r Ki hieu c5c phdn ttr cta hai tdp niry theo thd tu teng LArvr eAI teo pHUt. CAu 1 : Ta ndi sd trl nhi6n A ld mQt s6 "Pitago", n6u A la tdng binh phuong cira hai s6 H6y chrlng minh ding thtlc t{ nhi6n nAo d
a : { dl : 1, a2 = 2,... 0,,= n-} uit B = {b,, CD, AD, BC, ta cci h6 thtlc OHJ : BH . CH = *.7, br_t = n + 2,... bl = 2n] thi ta cci cdc n ER AE gi6 tri tuy6t d6i cria hi6u lo,- bil ld SuY ra -DF = ot-btl :2n-1 FC: al- bz: 2n - 3 Do dci didm M phAi nim tr6n do4n .EF. Dudng cao rlng vdi dinh M cia tan g;t6c ADM cd d6 dli ld .R, vi canh driy lb.2R, suy ra di6n . la,,-t- b,r-1! = 3 I a,,- b,rl : 1 tich tam gi6c ndy li R2. Do diOn tich cdc tam Do dd tdng c6c gi6 tri ndy s6 id : g16c ADM vd tsCM bang nhau. n6n trong (2n - 1) + (2n - 3) +... * 3 * I = n2 trt{dng hop c6c gcic B vd C vu6ng, chring ta Bay gid bang c6ch tSng c6c gre tri cta o, vi giAnr c6c giri tri cri;a b,(m6i brrdc thay ddi so CAu 4 : Bing suy luAn don giAn, dO thdy n6u drroc goi la 'ddi ch6" cdc phdn trl ctia A vd B), ta ldy ra 25 vi6n bi thi sE phai co it nhdt 7 vi6n bi chring ta crf thd d6 dnng truy cAp moi khA nang cung miu (nguy6n li Dirichle, 6 x 4 = 24 < 25) phAn b6 xen ko cdc g1a tri cria a, vd, b, trong Tdng qurit hdn, cho n hQp chfa cic vi6n bi khoAng tr) I ddn 2n md vAn git nguy6n thtl td drroc t6 bdn mdu, mdi h6p cci kh6ng it hon la vi6n ctia chrlrng irong trlng tdp con. Chf it ld qud bi duoc t6 cung mOt mdu. Tr6n l6n cric vi6n bi trinh niy s6 chdm dft khi md cria n hQp nAy. H6i cdn phAi ldy it nhdt bao nhi6u A : {al = rL * 7, a2 - n * 2,,.., a,, : 2i} vi6n dd chic chin cd ft vi6n bi cirng rniu. Cin vd trA lbi ln phAi ldy run(h -1) + 1vi6n bi. jl B = {br:7,bnl - 2...,b2:n- 7,bl:n) CAu 5 : Fh6p so sdnh hai xAu A va B dd Bay gid ta xern cl5;c g1a tri tuy€t ddi ctra hi6u duoc xAu C cri thd vidt nhtr sau :A & B + C thay ddi nhrr thd nio m6i khi o, "ddi ch6" cho b; : (ot & b,, a, & b r,..., a,, & b,) Ndu trtrdc khi ddi ch6, o, = kp bt = ft + 1 thi v6i ph6p torin tr6n cdc sd 0 vi 1 dttoc thrrc sau khi ddi ch6 ta c
t 15 Trong chrrong trinh PT ta cd mQt rlng dgng cira ph6p tinh tich ph6n : Di€n ticit ttinhphang gi6i hqnbrii hai dxdng th.ang I x -' a, x = o ii db thi cila hai cho hdnt F - lt = f {x) !2 1;it = fz$) tien tuc bAn [a , b] sao f/x) > f2(x), udi m.oi x e [a ,b] duqc cho bdi cOng o thic h" S = J tf ,$\ -f .Lx)Jd.x kem Gidi tich 12. NgO - rG Thuc Lanh tran.g 128, Phan Dtc Chinlt trang 159, . Hirttt l Trdn Van Hoo bdng t37). Trong cac hi tlti tdt nghi4p PTTH cia nvibrt Ntnt truoc G Z ddy ltdu nhtr nant nito cung co dong todn ,FI rln"E duns bii to6n tinh di6n tich bang tich ndy. d day, bai b-a9 ndy d6.cApddn,n;9f phan nay de- clrtlng-lDlnh_nl.ot so Daf, F aif;g finl"c. Phrtong phdp nny ccj hi6u ,- mo [ay loga. luc doi vdi c6c bf,t dang ttrtlc co chua nalrl rG 11 e z Vi aq 1. Cir.o cd.c sd p > 7, q > 1 tiaru hQ udi ruhau u] bdi r''A:'' - z Tttd thi: udi nrci a, b duong,ta c6 pq* ab = S ,,U' dd,ng thic Young) -I a Giai.x6tdudngcongy - r 1. tr-Lv6ix > 0,p z Ta crine cd r = 14 - /. Ta thdv (hinh 2) tdng di6n tich E t
ln (z-1) .ry < ntnn-n * lc+ C6e tran thir vAn dung phtrong ph6p ndy dd chting rninh cdc bdt ding thfrc ; I +L = ln(nn +;e1 -n) 1. b(o + 1) < { *blrub udi a, b > 0 n+l h-r 2. (n + l)ab
brong dci [o] Id phdn nguy6n cria o vir {a) = a N6u ta lai thay ddi rnQt phdn quy tic Josbphe Nhu vdy, cong thilc - foi-ta pitdn phan ciiio. hang dd cho c6c sd ddm ld 1,2,3,4 rrd didm cci sd truyto6ir d6i i6i Joslphe n > 4 lir ddm 4 mdi bi xda, cudi cung girl lai 3 didm kh6ng bi xcja thi cci thd goi s6 thf tu cria c6c .2*a,,-t\ ,2+bn-t, dldmtr;'li c6c sd siau Josizphe hang n vd ki du=l*"I * 1,b,,=l*"1 " i hi6u chung lb. u,,, ttu, ru,, (dinh nghia quy n4p Ndu d6t a, 0 va b, = I thi de ding kidm = cho cdc sd ndy arrqCtaul tudng tu nhu doi vdi tra lai rang c'Ong thrlc tr6n ci'n dfng cA v6i c6c s6 arr, bt,). Ndu dat u2 = 0, u2: 7, til' = 2 n=2vdn:3. thivdin>3tac6 Srl dung c6c c6ng thrlc ndy, ta l4p duoc bAng . 3*rr - r, , 3*'',,- , c'ic sd Josbphe tt tudng hang thdp l6n hang cao u,,=l*n{-#} ,u,,=7*"{ , j, d6i nhanh ch l. vd c5c s6 si6u JosdPhe. Sll dgng dign tieh ... -ffi ffi ffi > 8 Ddu"="co khiM ra trqng tam LABC 2) (Tidp theo trang Bang phrrong ph6p trinh biy tr6n. c6c ban ccj thd giAi c6c bdi t4p sau : AAt * MAt S, + S, MA S, * S-; --fuq- = s, - MAt = s, s; 1) Cho tam giSc d6u ABC, didm o d trong Trrong trl ta cci t MB sr + s1 *: ,, *- ., :itl.Ti:":$*';!^;il',u,;;",:'J};:ti;X; MBl= S' t_, ;; : AA'+ BB' +CC'kh6ng ddi. ", ";;- MC Z) Tr6n canh AC ctc MBC taldy didm D. :t, = * 1 *__, Chrlng t6 rang tdng cric b6" I*I dudng trbn Mq E ' S, n6i tidp cSrc tam g15c ABD vd BCD I6n hon brin MA MB +- +- MC kinh vbngtrbn n6i Li6p LABC' MAt ' MBt MCt = Al "' - 'Sr. , S, iJ, , Sl S.r. S, 3) Qua didnr M nim trong hinh binh hnnh :- l:!*Ij\' sr/ * l-:+-) \ s.r ' s,i , * Si-\/ 2 +/\ s.r ABCD tav6cscdrrdngthingP8vdQSsong \ s, song v6i cdc c4nh sc va AB. (CAc didnr P, Q, : > 2 +-2 +d 6. Ddu : -crt khi S, : S,:5.r. iR,SnantrOnAB, BC,CD,DA). Chtlngt6ring ttng vd bdt ding thrlc tr6n cric ditdng thing BS, PD vd MC d6ng quy' b) Nhan theo 3 ta duoc: "iit-""A* i* *.*.irs.+s.tr.q.+ (S,+S.,)(S.r+S1)(S1+S,) 4) Cho tam gi6c cdn ABC (AEt = ACl. Gqi CM --t, A1'81'cl lA chan c6cdtrdngphan giac tudng ,N ,N fr: "t'"2'*3 " ur-' ttng dinh A, B, C' Ki hi6u Sl = SM,B,{',' (n * b\z > 4ab "r. 'i ,AM,z ,BMr: ,CMr BC' - 1,n,il \r,*) \r+r) "- s = sMnc, ft = chrlng minh n6, t'av : 4S,S.1.4SJS7.'4SIS, S t- I rl - S,S,S.r - 5l 16
__r(_ .l D ^ 2 KET QUA TIII HOC SINH GIOI TOAN PTTH ToaN QUdC N,iM rroc r99s.t994 o ?liJTFrT,'lf3[i Dtlong, Htrg. tlidu_ (Nam Ha), Vn "'#i1""'ldlS"-"itbf'f t_h.i"g B 4rm tg9a. Cd-5t-d6i?d;'i Quang -liguvi+. HiSu- (Ninh Binh), Trdn Thantr' ga. ".".3 th-anfi p6o-"a ta"iiiiii sinh dy.tEi -o.bang A ra"6'f ilJ-y^qp&l3,lB,I,-P^I,6T,T,;*J $ffif fl s#fr #n:', lii{;*l *lrr' f,"{f sr#i &H Bdng B : $, $*i "'.tf 6 bai tdi da-cria--ca toan Ia 40 Ei8*. Xat !,i'd Tdng sd bii dtroc giAi la 12, chidm tilc lLlgs cuOc thi nhu sau : = 12,6% -l^0;r,)^Tdng sd bdi dtroc giai (dat tr) 20 didm trd Gidi nhdt: Kh6ng cci _cy5 cA hai bAhg?-, B-H 6b, .Gidi ryli : cd 2 giai tt,: 29,b didm ddn B0 601327 = 18,3Vc,. "hi6hi'iiic ^. glom) : HoAnA Vi6t- (Y6n Bzii); NCuydn XuAn Bdng A: Hirng (Dic Lfc) . ,ldfrg sd bai dtroc giAi la 48, chidm ti 16 Gidi ba; cd 6 giAi (tt 28 didm ddn 25 didm) : ^ 481232-= 2O,6Vo - _ T.dn Tuong Qudc, L6 BAo Lons. Nsuv6n Gidi nhd.t: cri 1 oi.Ai (3.9,b didnr) : Trdn Ngoc Minh rha4h tTay Niqh2 r,o FlonE'(od"1,6"i, Nam (Trtrdng DHfH HN1- -'-"' B*i.Q,-e"g Nam rLam OOnE), Vo fiBah-g:Ir""g Gidi ruhi; giei (tt (Trd VinE) J
- To6n hgo va dbi s6,s f,DHCp non K&$ffi& rH;6$ ffinnfr&ffi H"rrroilG CO tfrdnoi rdtnhidu nhrlnggl mAtathdyxungquanh nhrlng ve d-ei'xi tno .A ta nhjn ngini iang ngAy, lA nhttng kdt-quA c0a lN;ii inr'ai ni.n noc' c,ia'inien nnlen. Kh6ns phai nsAu nhi0n rna mOt c6i i6 ciy, mOt quA thdng, mOt con c6, mOt con rUa lf,dn,: f,rn lai c6 mOt hi nh dan3 ch-ung; vir mOt cAi v6 dc xinh ,i. *.s mQt chu6i thi0n hl kndng td xa ldc xa ld.lai duoc "ai cirnq m0t khuOn m6u hinh*hqc: chting co h)nh d4ng cdu tao theo .Jo nntrng,ltlon! xoSn 5e. ru 6i m0t sinh vat - m0t c'Ai cAyT0t Oang noalmOt cliri cdn -trirng v- v.. - tr.I n6 dd lA m0t bdi hec hlnh frocitrloi dang hoAn chinh nh6t Thi6n nhi6n dd s6ng tao ra v0 tru th.eo mOt ch!'dng tr)nh todn hec xAc di nh va ng,.roi ta nhAn ra ring cac pntldng ph-ep mir thr0n nhi0n dd sJ clqng thr'tdng lA ddn-gi an va trUc tiep' bit Od, tr.t mOt sd luong nhd cac phdn-ttt' thi6n rrhi6n xAy ;;;" ;4" ;"c cAu t'tic cJa tno tnci vat ti. rdi ctrng voi cech in-;""tiet kidm cac plrudng ti en'rhu v$y, chi-udng rr'at vai.hinh nlnn nq" cd bAn, thi6n nhi6n de tao cho i:h0ng cdu tr(c dycAc hinh dAnq bi4n fr6a v6 ctrng. N gudi t:r dA.phAt hi6n ra rdng hinh dang cd ie pi-l.i" tap nhdt cring chi ld nhr-lng "g"v;a ";r,rg /9 _/, I *,,/ ./ / / ,t / /Y't,','tl ,/ ,'tl, .k, L./ I t.H l/ Y-/ ) \ \__, \\s--:-/ i\j--r 6 la hlnh trin, hlnh t:rn o1 66 irrnh rd nQp ciia nhr.tng hlnh hlnh hoc c{, bAn. cl-)n g-i dn, guen thuOc.nh6t.,D ti"o niidu oach vu0ng. CAchiirh naycO ifr d"o-sanh vdi cicch-rt c6i a, b, c,,., mAng,Jai;"i;n'g'idftdttrOp i